Contact structures and Floer homology on 3-manifolds with boundary

带边界的 3 流形上的接触结构和 Floer 同源性

• 批准号: 1506157
• 负责人: Peter Ozsvath
• 金额: $ 15.95万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1506157&HistoricalAwards=false
• 关键词: Contact; structures; Floer; homology; manifolds

The principal investigator aims to understand the interplay between contact geometry in 3 dimensions and invariants of 3-manifolds. Contact structures are geometric objects which originated in the study of optics in the 19th century; in recent years they have become important tools in low-dimensional topology, providing information about the shapes of 3- and 4-dimensional spaces such as our universe, and in knot theory, which has seen emerging links to such topics as protein folding. Many of the tools which the PI uses to study such spaces, including Floer homology theories, draw heavily from mathematical physics, and they contain contact-geometric data which has provided a wealth of information about both the contact structures themselves and about the spaces in which they reside. The proposed research would investigate such data and uncover connections which it provides between disparate areas of mathematics, including algebra, topology, and dynamics.The PI intends to study invariants of contact structures and 3-manifolds, especially homological invariants of 3-manifolds with boundary, which arise from gauge theory and symplectic geometry. The first goal of this project is to develop applications to topology and to symplectic geometry of contact invariants in several Floer homology theories for sutured 3-manifolds. These potential applications include computable obstructions to Lagrangian concordances between Legendrian knots; bounds on the number of Reeb orbits in a contact 3-manifold, generalizing the proof of the Weinstein conjecture in this setting; and an intriguing conjecture relating Stein fillings of a 3-manifold to representations of its fundamental group. The second goal is to establish a relationship between different sutured Floer homology theories, whose corresponding closed 3-manifold invariants (Heegaard Floer homology, monopole Floer homology, and embedded contact homology) are now all known to be isomorphic, and to identify their respective contact invariants as well. The third goal is to investigate an emerging connection between Legendrian contact homology (LCH) and new sheaf-theoretic Legendrian knot invariants, and in doing so to apply algebro-geometric techniques to problems in contact geometry and hopefully understand the relationship of LCH to classical knot invariants.

主要研究者的目的是了解接触几何在3维和3流形的不变量之间的相互作用。 接触结构是一种几何对象，起源于19世纪的光学研究;近年来，它们已成为低维拓扑学的重要工具，提供了关于三维和四维空间（如我们的宇宙）形状的信息，并在纽结理论中，它已经与蛋白质折叠等主题建立了联系。 PI用来研究这种空间的许多工具，包括Floer同调理论，都大量来自数学物理，它们包含接触几何数据，这些数据提供了关于接触结构本身和它们所处空间的丰富信息。 拟议的研究将调查此类数据，并揭示它在代数、拓扑和动力学等不同数学领域之间提供的联系。PI打算研究接触结构和3-流形的不变量，特别是具有边界的3-流形的同调不变量，它源于规范理论和辛几何。 这个项目的第一个目标是发展应用拓扑和辛几何的接触不变量在几个弗洛尔同源理论缝合3流形。 这些潜在的应用包括可计算的障碍拉格朗日协调勒让德结之间;边界上的数量Reeb轨道在接触3流形，推广证明温斯坦猜想在这种情况下;和一个有趣的猜想有关Stein填充3流形表示其基本群。 第二个目标是建立不同的缝合弗洛尔同调理论之间的关系，其相应的封闭3-流形不变量（Heegaard弗洛尔同调，Heegaard弗洛尔同调和嵌入接触同调）现在都已知是同构的，并确定它们各自的接触不变量。 第三个目标是研究勒让德接触同调（LCH）和新的层理论勒让德结不变量之间的新兴联系，并在这样做时将代数几何技术应用于接触几何问题，并希望了解LCH与经典结不变量的关系。